Puzzle 1: Red to move and win
Solution

Red wants to take advantage of the awkward position of the black rook. If Black can set free the rook Red will be in trouble. Red wants to achieve a checkmate or transform the position into the won endgame Cannon and advisor vs. 2 advisors. The horse cannot win against 2 advisors, therefore 1.C2-1? P6.5 2.C2.5 P5+1 is a draw.

1. C2+8 K4+1
2. C2-6 K4-1
     
  2. ... R5.2
3. C2.6 A5+4
4. H6+8 mate
1 point for this sideline

2. ... A5+4
3. H6+4 K4-1
4. C2.6 A4-5
5. H4-5
yields a position of the main line.
3. C2.6
1 point
3. ... K4.5
  3. ... A5+4
4. H6+4 A4-5
5. H4-5 brings us to the main line (1 point)
4. H6+4 K5.4
5. H4-5 K4+1
  5. ... R5.2
6. H5+6 mate
6. C6-2
This move (2 points) brings Black into Zugzwang:
6. ... A5-4 7. H5-6 mate, 6. ... A5+6 7. H5-6 mate, 6. ... R5.1 7. H5+6 K4+1 8. A5+6 mate (Answer3, 1 point) 6. ... P6.5 7. A4+5 and Zugzwang is complete (Answer2, 1 point)
6. ... P6+1
After 6. ... P6.7 the game continues as in the main line, but Black won't even get the horse, e.g. 6. ... P6.7 7. H5-6 A5+4 8. H6-4 A4-5 9. H4-5 P7+1 10. H5+7 P7.8 11. C6-1 and Red wins (1 point).
7. H5-6 A5+4
8. H6-4 A4-5
9. H4-5 P6.5
2 points for the rest of the main line, Red will capture Black's pawn (e.g. as in Answer1), the resulting endgame cannon and advisor vs. 2 advisors is a book win.
Puzzle 2: Red to move and checkmate
Solution

This positions was seen with reversed colours in a Chinese master game in 1986. On a first glance Red seems to have some problems to release the pin on the 6th file. A closer look reveals that Black's defence collapses immediately if the only defending piece, the black cannon, is eliminated.

1. R4+3      
After this move (1 point) the black king is without defence against the combined attack of Red's cannon, horse and pawn.
1. ... A5+6
  1. ... K6.5
2. H8+7 mate
2. C5.4 A6-5
  2. ... K6.5
3. H8+7 K5+1
4. P5+1 K5.4
5. P7-6
With threats C4.6 mate and H6+8 mate
5. ... A4+5
6. C4.6 A5+4
7. H6+8 mate
1 point for this side line (Answer3)
3. P5.4 A5+6
  3. ... K6.5
4. H8+7 mate
4. P4+1 K6.5
5. H8+7 K5+1
6. H7-6 K5-1
  6. ... K5.4
7. C6.4 mate
7. P4+1
Threatening not only 8. H6+7 mate, but also 8. P4+1 K5.6 H6+4 mate (1 point for P4+1)
7. ... C5-8
  7. ... R7-1
    7. ... A4+5
8. P4.5 K5.4
      8. ... K5.6
9. H6+4 mate
    9. C4.6 mate
  8. K4-1 R7.3
9. P4+1 K5.6
10. H6+4 mate
1 point for the refutation of the alternatives to 7. ... C5-8 (Answer2)
8. C4.5 C5.4
9. H6+5 mate
1 point for the final moves of the main line (Answer1).
Puzzle 3: Red to move and checkmate
Solution

At first sight Red seems to be in trouble, Black has material advantage and a strong attack, Red cannot force a running mate. On the other hand, Black neither has a running mate, thus Red has the time to strengthen his attack with a quiet move.

1. P+.2      
An immediate 1.C1+7 would allow 1.... E7+9 2.P++1 K6+1 H4+2 K6+1 and Black's king has escaped. Therefore, Red imprisons the black elephant, prevents E7+9 and sets up the threat 2.C1+7 K6+1 3.H4+2 K6+1 4.C1-2 mate. 3 points for the key move 1.P+.2
1. ... A5+4
  1. ... K6.5
2. C1+7 A5-6
3. H4+6 K5+1
4. C1-1 mate
1 point for this side line (Answer5)

1. ... E5-3
2. C1+7 E7+5
3. P2+1 E5-7
    3. ... K6+1
4. H4+2 K6+1
5. C1-2 mate
  4. P2.3 K6+1
5. H4+2 K6+1
6. C1-2 mate
1 point for this side line (Answer4)

1. ... A5+6
2. C1+7 K6+1
3. H4+6 R1.2
    3. ... A6-5
4. P2.3 K6+1
5. H6-5 mate
1 point

3. ... K6.5
4. C1-1 mate
  4. P2.3 K6.5
5. C1-1 mate
1 point (Answer3)
2. C1+7 K6+1
3. P2.3 K6+1
4. H4-3 E5+7
4. ... E5+3 and 4. ... E5-3 can be treated in the same way, all other allow H3+2 mate or H3+5 mate.
5. H3+5 K6.5
6. C1-5
Threatening C1.5 mate
6. ... K5-1
7. C1+4 K5+1
  7. ... K5-1
8. H5+4 K5.6
9. C1+1 E7+9
10. H4+3 mate
1 point for this side line (Answer2)
8. H5+3 K5.6
9. C1-4
Black has no defence against the mate threat H3-5 K6.5 C1.5. He can only defer the end with P4+1 K5.6 R1.4 K6+1. 2 points for the final moves of the main line (Answer1).